It is interesting to explore any possible generalizations of Theorem B, in particular, to discuss whether or not (REF ) holds for \(L^q\)  differential form \(\omega \, ,\)  where \(q\ne 2\, .\)  Some study of this generalization can be found in [1]}, its counter-examples are given by D. Alexandru-Rugina (see [2]}) on the hyperbolic space \(H^m_{-1}, m\ge 3\, ,\)  and relevant remarks of Pigola-Rigoli-Setti are discussed in [3]}.
